Joint source and relay beamforming design and time allocation for wireless powered multi-relay multi-user network

Abstract

This paper studies joint source and relay beamforming optimization and time allocation for a wireless powered multi-relay multi-user network, where a multi-antenna base station (BS) transmits information to users with the assistance of multiple single-antenna relays. Considering that the relays are energy-constrained, which are powered by the BS wirelessly, we aim to maximize the minimum rate among the users by jointly optimizing the information and energy beamforming at the BS, the distributed beamforming at the relays, and the time allocation under the power constraints and the energy causality constraints at the BS and relays, respectively. In order to tackle the formulated optimization problem, we propose an iterative algorithm by alternatively solving two subproblems. Numerical results show that the proposed joint optimization scheme obviously outperforms other schemes.

Appendices

Appendices

1.1 Appendix A: Proof of Lemma 1

Because (SP1$''$) is convex and satisfies Slater’s condition, the KKT conditions of (SP1$''$) are sufficient and necessary conditions for the optimal solution to (SP1$''$). Let $\{\chi _k\ge 0\}$, $\{\iota _{m}\ge 0\}$, $\delta \ge 0$, and $\{{\varvec{Q}}_k\succeq {\varvec{0}}\}$ denote the Lagrangian multipliers corresponding to (11b), (11c), the second constraint in (11d), and $\{{\varvec{X}}_{k}\succeq {\varvec{0}}\}$, respectively. From the KKT conditions of (SP1$''$) associated with $\{{\varvec{X}}_{k}\}$, we have

$$\begin{aligned}&\frac{\partial {\mathcal {L}}}{\partial {\varvec{X}}_{k}}=-{\varvec{Q}}_k+\kappa {\varvec{I}}_L-\frac{\chi _{k}(1-\tau ){\varvec{F}}_k}{\big ((1-\tau ){\tilde{\sigma }}^2+\mathrm {tr}({\varvec{F}}_k{\varvec{X}}_{k})\big )\ln 2}\nonumber \\&\quad \quad \quad +\sum _{m=1}^{M}\iota _{m}{\varvec{J}}_{k,m}+\delta {\varvec{I}}_L={\varvec{0}},~k\in \{1,2,\ldots ,K\}, \end{aligned}$$

(30a)

$$\begin{aligned}&\mathrm {tr}({\varvec{Q}}_k{\varvec{X}}_{k}) ={\varvec{0}},~k\in \{1,2,\ldots ,K\}, \end{aligned}$$

(30b)

where ${\mathcal {L}}$ is the Lagrangian function. Also, $(1-\tau )$ and $\{\mathrm {tr}({\varvec{F}}_k{\varvec{X}}_{k})\}$ are greater than zero. Then we rewrite (30a) into the following form.

$$\begin{aligned} {\varvec{Q}}_k=\varvec{\varPsi }_k-\chi _k\omega _k{\varvec{F}}_k,~\forall k, \end{aligned}$$

(31)

where $\omega _k\triangleq \frac{1-\tau }{\big ((1-\tau ){\tilde{\sigma }}^2+\mathrm {tr}({\varvec{F}}_k{\varvec{X}}_{k})\big )\ln 2}>0$ and $\varvec{\varPsi }_k\triangleq \kappa {\varvec{I}}_L+\delta {\varvec{I}}_L$ $+\sum _{m=1}^{M}\iota _{m}{\varvec{J}}_{k,m}$. Since ${\varvec{I}}_L$ is full-rank, $\{{\varvec{J}}_{k,m}\}$ are positive semidefinite diagonal matrices, and $\kappa >0$, one can obtain that $\{\varvec{\varPsi }_k\}$ are always full rank. Then from (31), we have that the rank of ${\varvec{Q}}_k~(\forall k)$ is $L{-}1$ or L due to $\mathrm {rank}({\varvec{F}}_k)=1$. From (30b), we have $\mathrm {rank}({\varvec{Q}}_k)+\mathrm {rank}({\varvec{X}}_{k})\le L$. Thus, one can obtain that $\mathrm {rank}({\varvec{X}}_{k})=1$ must be satisfied with the KKT conditions of (SP1$''$) [33]. It means that the optimal $\{{\varvec{X}}_{k}\}$ to (SP1$''$) are always rank-one.

1.2 Appendix B: Proof of Lemma 2

Since $\{{\varvec{U}}_{k,m}\}$ are positive semidefinite real-valued diagonal matrices, one can define ${\varvec{V}}_{k,m}\triangleq ({\varvec{U}}_{k,m})^{1/2}$. Then for $\{\mu _k>0\}$, we have

$$\begin{aligned} \frac{{\tilde{\varvec{\alpha }}}^{T}_{k}{\varvec{U}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}}{\mu _{k}}&=\frac{({\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k})^{T}({\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k})}{\mu _k}\nonumber \\ {}&=\mu _k\left( \frac{{\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}}{\mu _k}\right) ^{T}\frac{{\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}}{\mu _k},~\forall k,m. \end{aligned}$$

(32)

According to the definition of the perspective of a function as shown in [34, Sec. 3.2.6], one can see that the function $\mu _k\Big (\frac{{\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}}{\mu _k}\Big )^{T}\frac{{\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}}{\mu _k}$ is the perspective of the function $({\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}){^{T}}{\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}$. Since $({\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}){^{T}}{\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}$ is convex, one can prove that $\frac{{\tilde{\varvec{\alpha }}}^{T}_{k}{\varvec{U}}_{k,m}{\tilde{\varvec{\alpha }}}_{k}}{\mu _{k}}$ is jointly convex in $({\varvec{V}}_{k,m}{\tilde{\varvec{\alpha }}}_{k},\mu _k)$ for $\mu _{k}>0$.